1057: principles of communication system (i)twins.ee.nctu.edu.tw/courses/commu_07/lecture 03a-...

Commun. I Lecture3 - Basic Modulation Techniques ([email protected])

3-I-1

1057: Principles of Communication System (I)

1057: Principles of 1057: Principles of Communication System (I)Communication System (I)

Lecture 3Lecture 3--I Basic Modulation I Basic Modulation TechniquesTechniques


3-I-2

Modulation• A process to translate the information data, the message

signal, to a new spectral location depending on the intended frequency for transmission.

• Modulation techniques– The choice is influenced by the characteristics of the message

signal, the characteristics of transmitted channel, the desired performance, the use to be made o the transmitted data, and the economic factors,…

• Basic modulation techniques– Analog modulation

• Continuous-wave modulation– Linear Amplitude Modulation– Frequency Modulation– Phase Modulation

• Pulse modulation

m(t) xc(t)

modulated carrier


3-I-3

Linear and Angle Modulation• A general modulated carrier can be represented by

where ωc is carrier frequency• Once the carrier frequency is specified, only two parameters

are candidates to be varied in the modulation process– The instantaneous amplitude: A(t)– The instantaneous phase deviation: φ(t)

• When A(t) is linearly related to the modulating signal, it is called the linear modulation

• When φ(t) is linearly related to the modulating signal, it is called the frequency or phase modulation (or just the angle modulation)

)](cos[)()( tttAtx cc φω +=


3-I-4

Linear Double-Sideband (DSB) Modulation

• Definition:• Spectrum of a DSB signal

ttmAtx ccc ωcos)()( =

πω2

),(21)(

21)( c

cccccc fffMAffMAfX =−++=

fC

fC+W fC-W f

XC(f)

W -W f

M(f)

Upper sideband(USB)

Lower sideband(LSB)

ℑ

messagesignal

fC

fC+W fC-W f

XC(f)

Upper sideband(USB)

Lower sideband(LSB)


3-I-5

DSB Coherent Demodulation• Coherent (synchronous)

demodulator– The receiver side knows

(in advance) exactly the phase and frequency of the received signal

ttmAtmAtttmAttxtd

cCC

ccCcc

ωωωω

2cos)()( cos2]cos)([cos2)()(

+=⋅=⋅=

desired part High freq. noise

- What if the receiver reference is not coherent?


3-I-6

DSB ModulationPut it together…


3-I-7

Noise in Modulation Systems• Details in Chap. 6• A phase-locked loop (PLL) is considered in 3.4• A simplified analysis for a time-varying phase error

2cos(ωct+θ(t))

xr(t) d(t)

))(2cos()()(cos)( ))(cos(cos)(2)(

tttmAttmAttttmAtd

cCC

ccC

θωθθωω

++=+⋅=

1)(cos1 )(cos)()( ≤≤−= tttmtyD θθ

varying !!


3-I-8

A Phase Coherent Demodulation Carrier Generator

( )2

Narrowband BPF at 2fc

2÷f xr(t)

cos2ωct cosωct

ttmAtmAttmAtx cCCcCr ωω cos2)(21)(

21cos)()( 22222222 +==

DC

0 2fc -2fc

FT(m2(t))

Narrow BPF

f

Frequency divider

Carrier recovery circuit


3-I-9

Remarks• The spectrum of DSB signal does not contain a

discrete spectral component at the carrier frequency unless m(t) has a DC component.

• DSB systems with no carrier frequency component present are often referred to as suppressed carrier (SC) systems.

• If the carrier frequency is transmitted along with DSB signal, the demodulation process can be rather simplified.

• Alternatively, let’s see the following amplitude modulation (AM) scheme.


3-I-10

Amplitude Modulation (AM)• A DC bias A is added to m(t) prior to the modulation

process– The result is that a carrier component is present in the

transmitted signal• Definition

ttamAtAtmAtx

cnc

ccc

ωω

cos)](1[ cos)]([)(

+=

′+=


3-I-11

Remarks

• The parameter a is known as the modulation index• Coherent AM DSB demodulation

– precise, but it requires carrier recovery circuit• Incoherent detection

– Envelope detection

-fc fc 0 0

AM

ttamAtx cnCc ωcos)](1[)( +=


3-I-12

Envelope Detection

mn(t): the normalized message

A

tma t

)(min=

a: the modulation index

• The modulation index is defined such that if a=1, the minimum value of Ac[1+amn(t)] is zero– a <1, it results in Ac[1+amn(t)] >0 for all t

• All the information is just the envelop.• The envelop detection is a simple and straightforward

technique

ttamAtx cnCc ωcos)](1[)( +=

,)(min

)()(tm

tmtmt

n =

m(t): the original message

A: the DC bias


3-I-13

Remarks• The time constant RC of the envelop detector is

an important design parameter.• The appropriate RC time constant is related to

the carrier frequency fc and to the bandwidth W of the original signal m(t)– 1/fc << RC << 1/W, between then and must be well

separated from both


3-I-14

Power Efficiency of AM• Suppose that m(t) has zero mean, then the total power

contained in the AM modulator output is

• The power efficiency is defined by the power ratio of the input information to the transmitted signal

])([21

])()(2[21

2cos)()]([21)()]([

21

cos)()]([)(

222

222

2222

2222

tmAA

tmtmAAA

tAtmAAtmA

tAtmAtx

C

C

cCC

cCc

+′=

++′=

′++′+=

′+=

ω

ω

%100)(1

)(%100

)(

)(Efficiency

22

22

22

2

×+

=×+

≡≡tma

tma

tmA

tmE

n

n

⟨⋅⟩ denotes the time average value


3-I-15

Remarks• If the signal has symmetrical value, i.e. |minm(t)|=|maxm(t)|,

then |mn(t)|≤1 and hence ⟨mn2(t)⟩≤1.

– If a≤1, the maximum efficiency is 50%, e.g. the square wave-type

– For a sine wave, ⟨mn2(t)⟩=1/2, for a=1, the efficiency is 33.3%

– If we allow a>1,• Efficiency can exceed 50%, (a→∞, the efficiency=100%)• But, the envelope detector is precluded.

• The main advantage of AM is– A coherent reference is not necessary for demodulation as long

as a≤1• The disadvantage of AM

– The DC value of the message signal m(t) cannot be accurately recovered. (mixed with carrier)

%)100()(1

)(%)100(

)(

)(22

22

22

2

tma

tma

tmA

tmEfficiencyE

n

n

+=

+≡≡

Figure 3.4


3-I-16

Single-Sideband Modulation• In DSB, either the USB and the LSB have equal

amplitude and odd phase symmetry about the carrier frequency.– Each sideband contains sufficient information to

reconstruct m(t)– Bandwidth utilization is not efficient– Single-sideband (SSB) modulation is considered

• SSB modulation– A more complex signal processing technique (X)– It reduces the bandwidth of the modulator (O)– Two methods

• (Bandpass) Filtering easy to understand, but hard to implement• Hilbert transform or frequency transformation technique


3-I-17

Filtering SSB Modulation

Sideband filtering

• An ideal passband filter is necessary• The (very) low frequency component will be encapulated


3-I-18

Generation of Lower-Sideband Filter

• From the inverse Fourier transform…

)]sgn()[sgn(21)( ccL fffffH −−+=

)(2

)(2

)( cC

cC

DSB ffMAffMAfX −++=

)]sgn()()sgn()([4

)]()([4

)]sgn()()sgn()([41

)]sgn()()sgn()([41

)()()(

ccccC

ccC

ccccC

ccccC

LDSBc

ffffMffffMA

ffMffMA

ffffMffffMA

ffffMffffMA

fHfXfX

−−−+++

−++=

−−+−+−

+−+++=

⋅=

Xc(f)

XDSB(f)

ttmAffMffMAc

ccc

c ωcos)(2

)]}()([4

{1 =−++ℑ−

ttmAeejtmA

etmjetmjA

ffffMffffMA

cCtfjtfjC

tfjtfjC

ccccc

cc

cc

ωππ

ππ

sin)(ˆ2

)](21)[(ˆ

2

])(ˆ)(ˆ[4

)}sgn()()sgn()({4

22

22

1

=−=

−=

−−−++ℑ

−

−

−

Hilbert Transform


3-I-19

Time-Domain Representation of SSB Modulation

ttmAttmAtx cC

cC

c ωω sin)(ˆ2

cos)(2

)( +=

ttmAttmAtx cC

cC

c ωω sin)(ˆ2

cos)(2

)( −=


3-I-20

Hilbert Transform• Consider a filter that simply phase-shift all frequency

components of its input by -½π, that is

⎪⎩

⎪⎨

⎧

<−=>

=0,10,00,1

)sgn(fff

f

tth

π1)( =

fjfH sgn)( −=

t

h(t)

j

-j

f

H(f)

π/2

f

∠H(f)

-π/2

Not abs-integrable, since the value is infinite at t→0

⎩⎨⎧

<−>

=−

0,0,

)(fefe

fGf

f

α

α

αconsiderProof:

)sgn()(lim0

ffG =→ αα

∫ ∫∞

∞−

−−

+=−==ℑ

0

0

22221

)2(4);()]([

ttjdfeedfeetgfG ftjfftjf

παπα παπα

αthen

tj

tj

tgfG

ππαπ

α

α

ααα

=+

=

=ℑ

→

→

−

→

220

0

1

0

)2(4lim

);(lim)]([lim


3-I-21

Hilbert Transform

• Example 2.26

)(txt

thπ1)( = )(ˆ tx

∫∫∞

∞−

∞

∞−

−=

−= τ

πτττ

τπτ dtxd

txtx )(

)()()(ˆ


3-I-22

Properties of Hilbert Transform• The function is defined as the Hilbert transform of x(t),

then • The energy (or power) in a signal x(t) and its Hilbert

transform are equal.

• A signal and its Hilbert transform are orthogonal

• If c(t) and m(t) are signals with nonoverlapping spectra, where m(t) is lowpass and c(t) is highpass, then

)(ˆ tx)()(ˆ̂ txtx −=

)(ˆ tx2222222

)()()sgn()()()](ˆ[)(ˆ fXfXfjfXfHtxfX =−==ℑ=

∫∫ ∫

∞

∞−

∞

∞−

∞

∞−

==

=

0)()sgn(

)(ˆ)()(ˆ)(

2

*

dffXfj

dffXfXdttxtx

)]([)()]()([ tcHTtmtctmHT =

by Parseval’s Theorem

Check spectra density:


3-I-23

Proof:

∫ ∫∞

∞−

∞

∞−

′+

−−

′′=

ℑℑ=

fdfdefCfM

fCfMtctmtffj )(2

11

)()(

)]([)]([)()(π

where we assume that M(f)=0 for |f|>W and C(f’)=0 for |f’’|<WThen

)(ˆ)(

)]sgn()[()(

)]sgn()[()(

)]sgn()[()(

)]sgn()[()(

][)()()]()([

22

22

)(2

)(2

)(2

tctm

fdefjfCdfefM

fdfdefjfCefM

fdfdefjfCfM

fdfdeffjfCfM

fdfdeHTfCfMtctmHT

tfjftj

tfjftj

tffj

tffj

tffj

=

′′−′=

′′−′=

′′−′=

′′+−′=

′′=

∫ ∫∫ ∫∫ ∫∫ ∫∫ ∫

∞

∞−

∞

∞−

′

∞

∞−

∞

∞−

′

∞

∞−

∞

∞−

′+

∞

∞−

∞

∞−

′+

∞

∞−

∞

∞−

′+

ππ

ππ

π

π

π

By(2.282)

W

f+f ’=0

f

f ’

A

B

sgn(f+f ’) = 1

sgn(f+f ’) = -1

Example 2.27


3-I-24

Analytic Signal• Definition :

– for any real signal x(t), the analytic signal of x(t) is

• Then)(ˆ)()( txjtxtxp +=

⎩⎨⎧

<>

=

+=−+=

0,00),(2

]sgn1)[()](sgn[)()(

fffX

ffXffXjjfXfX p

)(ˆ)()( txjtxtxn −=

⎩⎨⎧

<>

=

−=

0),(20,0

]sgn1)[()(

ffXf

ffXfX n

Not even…


3-I-25

Equivalent Bandpass Signal Representation

• Bandpass signal:)(~ fX

B/2-B/2

2A

ff0

X(f)

-f0

( ))(2cos)()( 0 ttftatx θπ +=

( )

tftxtftxtfttatftta

ttftatx

IR 00

00

0

2sin)(2cos)( 2sin)(sin)(2cos)(cos)(

)(2cos)()(

πππθπθ

θπ

−≡−=

+=

ℑ

⎪⎩

⎪⎨

⎧

=

+=− )

)()((tan)(

)()()(1

22

txtxt

txtxta

R

I

IR

θwhere

inphase quadrature

)( of envelopcomplex thecalled is )(~ where})(~Re{})(Re{)( 00 2)(2

txtxetxetatx tfjttfj πθπ == +

A

carrier

zero-frequency parts of x(t), lowpass signals

complex envelop

)()()()(~ )( tjxtxetatx IRtj +== θ


3-I-26You may check the other direction !!

Remarks• The complex envelope of an arbitrary bandpass signal is not

a real existing signal. It is just an equivalent expression.

)()()(~)()(

)(ˆ)(

Re

02

02

txtxtxtxtx

txjtxP

e

e

I

R

tfj

tfj

⎯⎯⎯ ⎯←⎯⎯⎯ →⎯

⎯⎯ ⎯←⎯⎯ →⎯

↔+− π

π

tftxtftxtftxtftxtxtftxtftxtx

IRIR

IR

0000

00

2cos)(2sin)()2cos)((2sin)()(ˆ2sin)(2cos)()(

ππππππ

+=−−=−=

tfjtfjIR

tfjI

tfjR

IRP

etxetjxtxetjxetx

tftfjtxtfjtftxtxjtxtx0000 2222

0000

)(~)]()([)()(

]2sin2cos)[(]2sin2)[cos()(ˆ)()(ππππ

ππππ

=+=+=

−++=+=

tfjP etxtx 02)()(~ π−=

Proof:


3-I-27

Bandpass Systems• Physical

• Equivalent lowpass system

h(t)input

BP output

Bandpass

input Complex envelopoutput

Complex envelope

real(real BP)

(complex envelope)

)(tx

)(~ th

)()()( thtxty ∗=

)(~ tx)(~)(~)(~2 txthty ∗=

tfjtfj

tfjtfjtfj

etheth

ethethethth

00

000

2*2

*222

)(~21)(~

21

])(~[21)(~

21})(~Re{)(

ππ

πππ

−+=

+==

tfjtfjtfj etxetxetxtx 000 2*22 )(~21)(~

21})(~Re{)( πππ −+==

Note:


3-I-28

∫∫

∫∫

∫

∫

∞

∞−

−∞

∞−

−

∞

∞−

−∞

∞−

−−−∞

∞−

−

∞

∞−

−+−+

−+−=

⎟⎠⎞

⎜⎝⎛ −+−⎟⎠⎞

⎜⎝⎛ +=

−=∗=

λλλλλλ

λλλλλλ

λλλλλ

λλλ

λππλππ

ππ

λπλπλπλπ

detxhedetxhe

dtxhedtxhe

detxetxeheh

dtxhthtxty

fjtfjfjtfj

tfjtfj

tfjtfjfjfj

0000

00

0000

4*24*2

**22

)(2*)(22*2

)(~)(~41)(~)(~

41

)(~)(~41)(~)(~

41

)(~21)(~

21)(~

21)(~

21

)()()()()(

0)}(~)2(~{)(~)(~)(~)(~ *0

1*44* 00 =−−ℑ=∗=− −∞

∞−∫ fXffHtxethdetxh tfjfj πλπ λλλQ

( ) ( )( ){ }

})(~Re{

)(~)(~Re21

)(~)(~41)(~)(~

41

)(~)(~41)(~)(~

41)(

0

0

00

00

2

2

**22

**22

tfj

tfj

tfjtfj

tfjtfj

ety

etxth

txthetxthe

dtxhedtxhety

π

π

ππ

ππ λλλλλλ

=

∗=

∗+∗=

−+−=

−

∞

∞−

−∞

∞− ∫∫

Example 2.29


3-I-29

Physical Implementation of BP System

hR(t)

hI(t)

hI(t)

hR(t)

+

-

2yR(t)

+

+

2yI(t)

xR(t)

xI(t)

)(~)(~)(~2 txthty ∗=

tftytftytytftxtftxtxtfthtfthth

IR

IRIR

00

0000

2sin)(2cos)()(2sin)(2cos)()( ,2sin)(2cos)()(

ππππππ

−=−=−=


3-I-30

From Analytic Signal to SSB Signal• Mp(f): the positive-

frequency portion of M(f)

• Mn(f): the negative-frequency poriton of M(f)

• Apply the frequency-translation theorem to both the Mp(f) and Mn(f)– We obtain the upper-

sideband SSB signal and lower-sideband SSB signal, respectively


3-I-31

Coherent SSB Demodulation• Coherent detection with possibly phase error θ(t)

LPF

Kcos(ωct+θ(t))

d(t) xc(t) yd(t)

fc -fc

f

Xc(f)

2fc -2fc

f

LPF tfK 02cos π×

( )

)](2sin[)(ˆ)](2cos[)()(sin)(ˆ)(cos)(

))(cos(4sin)(ˆcos)(2

))(cos(4)()(

tttmAtttmAttmAttmA

ttttmttmAtttxtd

cCcCCC

cccC

cc

θωθωθθ

θωωωθω

+±++=

+×±=+×=

m

)(sin)(ˆ)(cos)()( ttmttmtyD θθ m=

if K=4,

message crosstalk


3-I-32

Remarks• If θ(t)≠0, the first term is a time-varying

attenuation of the message signal; while the second term is crosstalk.

• In general, there exist both frequency error Δfand phase error θ(t) in the local carrier, then– For LSB:

– For USB:( ) ( ))(2sin)(ˆ)(2cos)()( tfttmtfttmtyD θπθπ +Δ−+Δ=

( ) ( ))(2sin)(ˆ)(2cos)()( tftttmtfttmtyD θπθπ +Δ++Δ=


3-I-33

Carrier Insertion with Envelope Detector

( )

ttbtta

ttmAtKtmA

tKttmttmAte

cc

cC

cC

cccC

ωω

ωω

ωωω

sin)(cos)(

sin)(ˆ2

cos)(2

cossin)(ˆcos)(2

)(

±≡

±⎟⎠⎞

⎜⎝⎛ +=

+±=

• A complicated envelope detector is necessary

⎪⎩

⎪⎨

⎧

=

+=

−

)()(tan)(

)()()(

1

22

tatbt

tbtatR

θdefine

⎩⎨⎧

==

)(sin)()()(cos)()(

ttRtbttRta

θθ

that is,

22

)(ˆ2

)(2

)()( ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ +== tmAktmAtRty CC

D ktmAty CD +≈ )(

2)(

K large enough


3-I-34

Example 3.2•

⎩⎨⎧

+−=+−=

)3sin(9.0)2sin(4.0)sin()(ˆ)3cos(9.0)2cos(4.0)cos()(

111

111

ttttmttttm

ωωωωωω

)](cos[)(]sin)(ˆcos)([2

)( tttRttmttmAtx cccC

c θωωω +=±=

⎪⎪⎩

⎪⎪⎨

⎧

±=

+=

−

)()(ˆ

tan)(

)(ˆ)(2

)(

1

22

tmtmt

tmtmAtR C

θ

})()(ˆ

{tan))(( 1

tmtm

dtdtt

dtd

cc−±=+ ωθω

The instantaneous frequency of xc(t)


3-I-35

Remarks• Problems of SSB modulation

– The ideal (sharp cutoff) sideband filter– The poor low-frequency response

• Vestigial-sideband (VSB) filter– A filter that is mean-shifted conjugate anti-symmetric about the

carrier fc– An example:

• Let Hβ(f) be an LP anti-symmetric filter• Define the VSB filter H(f)

⎩⎨⎧

<++−−>−−−

=0),()(0),()(

)(fffHffUfffHffU

fHcc

cc

β

β

Hβ(f)

-1/2

1/2

β -β f

Hβ(f) = - Hβ(-f) ; Hβ(f) = 0 for |f| > β.

H(f)

1/2 U(f-fc)

fc+β f

fc-β fc

• Two advantages of VSB filter– The design of sideband filter is simplified– The low-frequency response is improved


3-I-36

Vestigial-Sideband Modulation•

f fc -fc

M(f-fc) M(f+fc)

f

M(f)

f fc -fc

M(f-fc)⋅H(f) M(f+fc)⋅H(f)

f

DSB

VSB

LPF

cos(ωct)

d(t)xc(t) yd(t)

( ) )()()()( fHffMffMfX ccc ⋅−++=

( ) ( ) )()2()(21)()()2(

21

)}({)(

cccc ffHffMfMffHfMffM

tdfD

−−+++++=

ℑ=

Demodulation

LPFafter )]()([)(

21)( ccD ffHffHfMfY −++=


3-I-37

Example

tBtA

tAte

cc

c

)cos(21)cos()1(

21

)cos(21)(

22

1VSB

ωωωωε

ωωε

+++−+

+=

tBtAtm 21 coscos)( ωω +=

tBtB

tAtAte

cc

cc

)cos(21)cos(

21

)cos(21)cos(

21)(

22

11DBS

ωωωω

ωωωω

−+++

−++=

The bandwidth required for VSB over the required for SSB/DSB is slightly increased by an offset


3-I-38

Frequency Translation and Mixing• The process of multiplying a message by a

periodic signal is called mixing.– The mixing effect: frequency translation– Example: A mixer

ttmttmte )2cos()(cos)()( 212 ωωω ±+=

The undesired term is removed by filter

The problem of the mixer:signal at the image frequency

ttk )2cos()( 21 ωω ±Let’s see the input signal:


3-I-39

Image Frequency

The image frequency must be eliminated !!


3-I-40

Image Frequency in Superheterodyne Receiver

• The fact is that it is hard to built a narrow bandpass filter at high frequency• The superheterodyne receiver has two amplification and filtering sections

prior to demodulation– A tunable RF filter followed by a fixed IF filter

• If we are attempting to receive a signal having carrier frequency ωc, we will also receive a signal at ωc+2ωIF if the local frequency is ωc+ωIF (or receive a signal at ωc−2ωIF if the local frequency is ωc−ωIF )

• The image frequency can be eliminated by the RF filter, which needs not be narrowband.

Tunable

Fixed

The mixer translates the input frequency ωc to the IF frequency ωIF


3-I-41

Remarks

• At the antenna: the desired signal with carrier ωc• After the RF filter: only the desired signal at ωc can go through (a wide BPF)• After the mixer: the desired signal at ωIF• After the IF filter: only the desired signal at ωIF can go through (a narrow

BPF) • The IF frequency is almost fixed• Two choices:

– Low-side tuning– High-side tuning


3-I-42

Smaller tuning range of LO is preferred.


3-I-43

Angle Modulation• The information is embedded in the angle (either in phase

or in frequency) of the sinusoid– Constant amplitude– General form:

– The phase deviation:

– The frequency deviation:

• Phase modulation (PM):

• Frequency modulation (FM):

dttd )(φ

)](cos[)( ttAtx cCc φω +=

)(tφ

)()( tmKt P ⋅=φ

)()( tmKdt

tdF ⋅=

φ

0000

)(2)()( φααπφααφ +=+= ∫∫t

td

t

tF dmfdmKt

Kp: deviation constant

fd: freq. deviation constant

)](cos[)( tmKtAtx PcCc += ω

])(2cos[)(0∫+=

t

tdcCc dmftAtx ααπω


3-I-44

Example

Phase changes

frequency changes

)](cos[)( tmKtAtx PcCc += ω

])(2cos[)(0∫+=

t

tdcCc dmftAtx ααπω

PM

FM


3-I-45

NarrowBand Angle Modulation•

ttAtAetjAeA

tjeA

tjtjeA

eeAttAtx

cCcCtj

Ctj

C

tjC

tjC

tjtjCcCc

cc

c

c

c

ωφωφ

φ

φφ

φω

ωω

ω

ω

φω

sin)(cos])(Re[

)]}(1[Re{

!2))(()(1Re

}Re{)](cos[)(2

)(

−=+=

+≈⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+++=

=+=

L

If |φ(t)| is much less than unity

Tayler series expansion..

carrier message×sinωc


3-I-46

If the input is m(t)=Acosωmt

The carrier and the resultant of the sidebands for narrowband angle modulation with sinusoidal modulation are in phase quadrature.


3-I-47

Frequency-Domain Behavior of Angle Modulation

• Consider the sinusoidal signal– Note that, any message signal can be decomposed into

combinations of sinusoids– WLOG, we may ssume that φ(t)=βsinωmt, and xc (t)=

Accos(ωct+βsinωmt), then

– Then

}Re{)( sin tjtjcc

mc eeAtx ωβω=

Periodic with period 2π/ωm

)(21

21

)sin(

sin

0

sin

0

0

βπ

πω

π

π

β

ωπ

ωπ

ωωβωωβ

nxnxj

tjntjmT tjntjn

Jdxe

dteedteeT

C m

m

mmmm

≡=

==

∫

∫∫

−

−−

−

−−

Its Fourier series coefficients

Bessel function of the 1st kind of order n and argument β

∑∞

−∞=

⋅=n

tjnn

tj mm eJe ωωβ β )(sinby Fourier series

∑∑∞

−∞=

∞

−∞=

+==n

mcnCn

tjnn

tjCc tnJAeJeAtx mc )cos()(})(Re{)( ωωββ ωω

∑∑∞

−∞=

∞

−∞=

+++−−=n

cmncn

cmncc nJAnJAfX )()()()()( ωωωδβωωωδβ


3-I-48

Frequency-Domain Behavior of Angle Modulation

The spectrum has components at the carrier frequency and has an infinite number of sidebands separated from the carrier frequency by integer multipliers of modulation frequency ωm


3-I-49

About the Bessel Function Jn(β)• Jn(β) is real-valued function•

•

• When β is small (β<<1) narrowband Angle Modulation

•

⎩⎨⎧

−==

−

−

oddn ),()(evenn ),()(

ββββ

nn

nn

JJJJ

)()(2)(

)(2)()(

11

11

βββ

β

ββ

ββ

−+

+−

−=⇒

=+

nnn

nnn

JJnJ

JnJJ

(by definition)

(proved by induction)

2for 0)( and ,2)( ,1)( 10 ≥≈≈≈ nJJJ n ββββ

1)(2 =∑∞

−∞=nnJ β Note: for angle modulation, we have

carrier nulls: the β values s.t. Jn(β) = 0

Table3.2


3-I-50

Remarks• The above analysis is based on the assumption that

φ(t)=βsinωmt, and we did not specify the modulator type– PM this means that m(t)=Asinωmt and β=kpA

– FM this means that m(t)=Acosωmt and β=2πfdA/ωm=fdA/fm

• Power in Angle modulated signal∫=

t

d dmft ααπφ )(2)(

( ) ( ) 22222

21sin2cos

21

21)sincos()( CmcCCmcCc AttAAttAtx =++=+= ωβωωβω

Constant transmitter power !!


3-I-51

Bandwidth of Angle Modulation Signal

• For large n, , hence, for fixed β,

• The bandwidth of an angle modulation signal is infinite, strictly speaking.

• Define the power ratio Pr

• Since the values of Jn(β) become negligible for sufficient large n, the bandwidth of an angle modulation signal can then be determined by an acceptable power ratio, says k, then

!2)(

nJ n

n

nββ ≈ 0)(lim =

∞→βnn

J

∑∑

=

−= +===k

nn

C

k

knnC

r JJA

JAkP

1

220

2

22

)(2)(

21

)(21

power totalpower components ββ

β

98.0for ,)1(22 ≥+≈= rmm PfkfBW β


3-I-52

General Case• For arbitrary m(t), a general accepted expression for BW is

through the definition ratio D

• For PM– Special case

• For FM– Special case

• The bandwidth is BW≈2(D+1)W

)(

)(max21

)( ofBW deviationfrequency Peak

tm

t

BWdt

td

tmD

φπ=≡

)](cos[)( ttAtx cc φω +=

)()( tmkt P=φtAtm mωsin)( = ,)(max mPt

AKdt

td ωφ=

)()( tmkdt

tdf=

φ))(max(

)(

tmBW

fDt

tm

d=

,sin)( tAtm mω=

AkD p=

m

d

fAfD =

Carson’s rule

1. D<<1, BW≈2W, narrowband angle modulated signal2. D>>1, BW≈2DW=2fd[max|m(t)|] , wideband angle modulated signal

D plays the same role as βfor sinusoidal signal


3-I-53

Example 3.6


3-I-54

Narrowband-to-Wideband Conversion

• Indirect Frequency Modulation--Two stages for generating wideband FM– Narrowband FM– Frequency multiplier

• The mixer output• After BPF

))(cos()( 0 ttAtx C φω +=

))(cos()( 0 tntnAty C φω +=tte LOLO ωcos2)( =

)]()cos[()]()cos[()( 00 tntnAtntnAte LOCLOC φωωφωω +−+++=

LOc

LOc

nn

ωωωωωω

−=+=

0

0 or ,

))(cos()( tntAtx cC φω +=Note: one can use Carson’s rule to determine the BW of the BPF if the transmitted signal is to contain 98% of the power in xc(t)


3-I-55

• Direct Frequency Modulation: voltage-controlled-oscillator (VCO)• This circuit has an oscillation frequency when x(t) slowly varies

∫+≈t

ccc dxfCCtft ααππθ )(

222)(

0

Narrowband-to-Wideband Conversion

)(cos)( tAtx ccc θ=


3-I-56

Demodulation of Angle Modulated Signal

• Frequency discriminator– A device that yields an output proportional to the

frequency deviation of the input– Received signal– The output of an ideal discriminator

– For FM

– For PM: Integration of the discriminator output yields a signal proportional to m(t)

• FM discriminator followed by an integrator

))(cos()( ttAtx ccr φω +=

dttdKty DD)(

21)( φπ

= KD: discriminator constant

,)(2)( ∫=t

d dmft ααπφ )()( tmfKty dDD =

linear

)()( tmkKty pDD =


3-I-57

FM Discriminator

•

• The output of envelope detector

• To reduce the channel noise effect, one applies the following

)](cos[)( ttAtx ccr φω +=

)](sin[)()()( ttdt

tdAdt

tdxte ccCr φωφω +⎟

⎠⎞

⎜⎝⎛ +−==

⎟⎠⎞

⎜⎝⎛ +=

dttdAty cc)()( φω This is always positive, if t

dttd

c ∀−> ,)(φω

CDdCcD AKtmfAdt

tdAty ππφ 2)(2)()( =→==


3-I-58

Ideal Differentiator•

dtd xr(t) e(t)

H(f)=j2πf

E(f) Xr(f)

H(f)

f

j


3-I-59

Realization of Discriminator (1)• Time delay block

• RC network

)()()( τ−−= txtxte rr

dttdxte r )()(lim

0=

→ ττ

dttdxte r )()( τ≈

fRCjfRCj

fCjR

RfHπ

π

π21

2

21)(

+=

+=

12 if ,2)( <<≈ fRCfRCjfH ππ

RCAK CD π2=

A highpass filter

For f=10MHz, KD is very small


3-I-60

Realization of Discriminator (2)• By using BPF

– Linear region ≈ differentiator– Disadvantages

• Small linear region• DC bias (H(f) should be 0 at fc)

• Balanced Discriminator – Advantages

• Wider linear range• No DC bias

f

|H(f)|

fc

H1(f): BPF at fc+Δ

H2(f): BPF at fc-Δ

Envelope detector

Envelope detector

xr(t)

y1(t)

y2(t)

yD(t) +

-

)()()( 21 fHfHfH −=


3-I-61

1057: principles of communication system (i)twins.ee.nctu.edu.tw/courses/commu_07/lecture 03a-...

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